Digital Quantum Simulations of the Non-Resonant Open Tavis-Cummings Model

The open Tavis–Cummings model consists of $N$ quantum emitters interacting with a common cavity mode, accounts for losses and decoherence, and is frequently explored for quantum information processing and designing quantum devices. As $N$ increases, it becomes harder to simulate the open Tavis–Cummings model using traditional methods. To address this problem, we implement two quantum algorithms for simulating the dynamics of this model in the inhomogeneous, non-resonant regime, with up to three excitations in the cavity. We show that the implemented algorithms have gate complexities that scale polynomially, as $O(N^2)$ and $O(N^3)$, while the number of qubits used by these algorithms (space complexity) scales linearly as $O(N)$. One of these algorithms is the sampling-based wave matrix Lindbladization algorithm, for which we propose two protocols to implement its system-independent fixed interaction, resolving key open questions of [Patel and Wilde, Open Sys. & Info. Dyn., 30:2350014 (2023)]. We benchmark our results against a classical differential equation solver in a variety of scenarios and demonstrate that our algorithms accurately reproduce the expected dynamics.

💡 Research Summary

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The paper tackles the challenging problem of simulating the open Tavis‑Cummings (TC) model—a system of N two‑level quantum emitters collectively coupled to a single cavity mode—under realistic conditions that include non‑resonant detuning, inhomogeneous coupling strengths, cavity and emitter losses, and an external coherent pump. Classical approaches based on vectorizing the Lindblad master equation scale exponentially in the number of emitters (the Liouville space has dimension 4ᴺ), making them infeasible for moderate N. To overcome this, the authors develop two digital quantum algorithms that achieve polynomial scaling in both gate count and qubit resources.

The first algorithm, called the Split J‑Matrix algorithm, builds on the earlier J‑Matrix method but introduces a more efficient decomposition of the Hamiltonian and Lindblad operators. The authors map the cavity, which is allowed to host up to three photons, onto two qubits (encoding 0‑3 excitations) and each emitter onto a single qubit. This encoding yields a total of O(N) qubits. The Hamiltonian and dissipators are expressed as sums of local terms; first‑order and second‑order Trotter‑Suzuki formulas are then applied to approximate the full time‑evolution operator. Each Trotter step requires only O(N²) one‑ and two‑qubit gates because each local term acts on a constant‑size subsystem. The space complexity remains linear, and the authors provide a detailed gate‑count analysis showing that the algorithm scales quadratically with N.

The second algorithm is the Wave‑Matrix Lindbladization (WML) method, originally proposed for open‑system simulation but previously lacking a concrete implementation of its system‑independent fixed interaction e_M^Δ. The authors resolve this by presenting two protocols. Protocol 1 uses the linear‑combination‑of‑unitaries (LCU) technique to synthesize e_M^Δ as a weighted sum of easily implementable unitaries. Protocol 2 exploits a channel‑exchange construction that achieves the same effect with a different gate structure. Both protocols assume that each Lindblad operator L_i is local (acts on a constant number of qubits) and can be prepared as a program state |ψ_L⟩. Under these assumptions, the overall gate complexity of the WML algorithm is O(N³), while the number of qubits remains O(N). The authors rigorously bound the sampling error and gate overhead in Theorems 1 and 2, and they provide proofs of the sample‑complexity and gate‑complexity results in the supplementary material.

To validate the algorithms, the authors benchmark them against a high‑precision classical differential‑equation solver across a variety of parameter regimes: homogeneous vs. inhomogeneous emitter frequencies, resonant vs. non‑resonant detuning, weak vs. strong coupling, and low vs. high loss rates (κ for the cavity, γ for the emitters). They compute two key observables: (i) the population of excitations in the cavity and each emitter, ⟨a†a⟩ and ⟨σ⁺_jσ⁻_j⟩, and (ii) the steady‑state second‑order photon correlation function g^{(2)}(0) = ⟨a†a†aa⟩ / ⟨a†a⟩². Across all tested scenarios, the quantum‑algorithm results match the classical solutions within the normalized diamond distance error bounds, confirming both accuracy and stability. Notably, the quantum simulations remain tractable even when the classical Liouville‑space approach would require prohibitive memory (e.g., N ≈ 10–20 emitters with three photon excitations).

The paper’s contributions are threefold: (1) it resolves an open implementation question for the WML algorithm by providing concrete, scalable protocols for the fixed interaction; (2) it demonstrates that both the Split J‑Matrix and WML algorithms achieve polynomial gate and linear qubit scaling, representing an exponential improvement over conventional classical solvers; and (3) it shows that these algorithms can handle the full non‑resonant, inhomogeneous, and lossy TC model, which is beyond the reach of many existing analytic or numerical techniques. The authors discuss extensions such as increasing the photon‑excitation cutoff by adding more cavity qubits, applying the methods to non‑Markovian or non‑linear open systems, and exploring error‑mitigation strategies for near‑term quantum hardware.

In summary, this work establishes a practical framework for digital quantum simulation of complex open quantum‑optical systems. By delivering concrete algorithms with provable resource bounds and demonstrating their performance on a physically rich model, the paper paves the way for future quantum‑computational studies of collective light‑matter interactions, quantum memories, and engineered photonic devices that rely on many‑body open‑system dynamics.